I am currently intending to analyse how the relative position of the rotor-stator system influence the flow in a turbine annulus (using Frozen Rotor). However, as I don't have much available time. I would like to know if it is at all possible to obtain this by re-positioning the mesh and using some clever commands.

As seen in the image below, this realigning would prevent me from changing the blade position in the rotor CAD model itself and re-meshing everything again...

Yes, this is possible. In fact for many cases it will be a very good approximation of the true flow and much quicker and easier to do than a full transient model.

Also, you will only need to mesh this once. In CFX-Pre you can transform the mesh to the new positions and re-run. And I think you can also parameterise it so you can run it as a single big job in workbench.

It is important to note that you can't use this Rotational Offset feature with the pitch change set to 'none', given it's not exactly aligned. The solution was to set it to value=1 (ie no pitch change).